A tribe on Mars called Upials uses the Marsupial plane M2 co

A tribe on Mars called Upials uses the Marsupial plane M^2 consisting of points (x, y, z) in the Cartesian space satisfying x^2 + y^2 + z^2 = 1. Marsupial lines in M^2 are defined as intersections of M^2 with Cartesian planes of the form a middot x + b middot y + c middot z = 0, where (a, b, c) notequalto (0, 0, 0). Show that for every two different points P and Q of M^2 there is a Marsupial line l containing them. Is it true that if two Marsupial lines l_1 and l_2 contain points P and Q, then l_1 = l_2 in P notequalto Q? Justify your answer. A tribe on Mars called Upials uses the Marsupial plane M^2 consisting of points (x, y, z) in the Cartesian space satisfying x^2 + y^2 + z^2 =1. Marsupial lines in M^2 are defined as intersections of M^2 with Cartesian planes of the form a middot x + b middot y + c middot z = 0, where (a, b, c) notequalto (0, 0, 0). Can you define the concept of Marsupial lines being parallel? Justify your answer.

Solution

Try to visualize this problem. The marsupial plane is nothing but a spherical plane of radius 1 unit (from origin) and when it is cut by a plane, the intersection will be a circle.

Thus the lines in marsupial plane will be circles in Euclidean plane.

Now take any point P(x1,y,1,z1) and Q(x2,y2,z2,)

Both points lie on the intersection between planes ax+by+cz=1 and the marsupial plane.

But the line of intersection of the plane ax+by+cz=1 and the marsupial plane is a circle.

The distance of each point on this circle will be equal to 1 as they lie on the marsupila plane (which is a spherial plane)

This will mean that point P and Q which lie on intersection of ax+by +cz=1 and marsupial plane will also lie on this circle (intersection) to satisfy the condition of being on the linear plane and on the marsupial plane.

Hence both P&Q will lie on the circle made by intersection ofp lanes. But the circle is nothing but a line in the frame of marsupial plane .Hence both P&Q will be colinear .

2. Consider that a given marsupial plane is cut by a laminar plane defined byP: ax+by+cz=1 . Clealry they will make a euclidean circle at intersection .

Now take another plane P1 parallel to plane P but which has higher coeffiecent of variables than P.

Clealry plane P1 will be farther away from P. Let P1 cut marsupial plane . Thus their intersection will be another circle. Clearly the radius of this circle will be less than the radius of circle cut by P.

However both the circles will be co axial . (why? because P and P1 are parallel to each other).

These euclidean circles will be lstraight lnies on marsupial plane and they will be parallel to eacho ther since they are axial.

Thus generallizing it we can say that two lines in a martian plane will be pralllel if they are cut by two planes which are parallel to each other.